Hi Jo
Working with a relocations/ removals organization is a good start to getting into Global Mobility. GM jobs are mostly in the East - Brisbane, Sydney and Melbourne so if you are based there, you are in luck. I'm in Perth and has been searching for GM jobs for the past couple of months.
HR qualifications will help in your career path. A GM specialist has to deal with tax, visa, comp &
ben etc...
If you are interested in having specialised qualifications in GM, take a look at this
Global Mobility Specialist (GMS?) Designation
The first step really for you is to go internal... i.e. to be the client.
Hope the above helps. Good luck and keep us updated.